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I'm having some problems finding the volume between a sphere of radius 5 and the cone $z = -\sqrt{x^2 + y^2}$.

the bounds I got for spherical coordinates are $0$ to $2\pi$ for $\theta$, $3\pi/4$ to $\pi$ for $\phi$, $0$ to $5$ for $r$, however I'm not sure if these are correct. My thought process is that if you let say $x$ or $y = 0$ in the equation for a cone, you get an inverted modulus graph which is where I got the bounds for $\phi$.

For cylindrical coordinates I got bounds of $0$ to $2\pi$ for $\theta$, $0$ to $5$ for $r$ and $0$ to $\sqrt{25-r^2}$ for $z$ and after evaluating the triple integral I got a volume of $125/3$.

Could someone run me through their thought process and show me where I went wrong?

Widawensen
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1 Answers1

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For spherical coordinates, your bounds look fine.

For cylindrical coordinates, though, you have $0\le r\le 5$; that is wrong. $r$ ranges from $0$ to the value that $r$ takes along the circle of intersection of the cone and the sphere.

I got $\frac{125}{3}(2-\sqrt{2})\pi$ for the value of the integral.

rogerl
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